Find dy/dx x^3+xy=e^y

Solution:

Step:1

Apply differentiation to both sides of the given equation $x^3 + xy = e^y$ with respect to $x$.

Step:2

Differentiate the left-hand side term by term.

Step:2.1

Take the derivative.

Step:2.1.1

Utilize the Sum Rule for differentiation: the derivative of a sum is the sum of the derivatives, $\frac{d}{dx}(x^3) + \frac{d}{dx}(xy)$.

Step:2.1.2

Apply the Power Rule, which says that the derivative of $x^n$ is $nx^{n-1}$, to get $3x^2 + \frac{d}{dx}(xy)$.

Step:2.2

Compute the derivative of the product $xy$.

Step:2.2.1

Use the Product Rule for differentiation: the derivative of the product of two functions is $f'(x)g(x) + f(x)g'(x)$, where $f(x) = x$ and $g(x) = y$. This yields $3x^2 + x\frac{d}{dx}(y) + y(1)$.

Step:2.2.2

Express $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$.

Step:2.2.3

Apply the Power Rule to $x$, which is $x^1$, to get $3x^2 + x\frac{dy}{dx} + y$.

Step:2.2.4

Multiply $y$ by 1 to maintain the expression $3x^2 + x\frac{dy}{dx} + y$.

Step:3

Differentiate the right-hand side of the equation.

Step:3.1

Employ the Chain Rule for differentiation, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Step:3.1.1

Set $u = y$ and differentiate $e^u$ with respect to $u$ and then multiply by $\frac{dy}{dx}$.

Step:3.1.2

Use the Exponential Rule, which states that the derivative of $e^u$ is $e^u \ln(e)$, simplifying to $e^u\frac{dy}{dx}$.

Step:3.1.3

Substitute $y$ back in for $u$ to get $e^y\frac{dy}{dx}$.

Step:4

Combine the differentiated left and right sides to form the equation $3x^2 + x\frac{dy}{dx} + y = e^y\frac{dy}{dx}$.

Step:5

Isolate $\frac{dy}{dx}$.

Step:5.1

Rearrange the term $e^y\frac{dy}{dx}$ to the left side to get $3x^2 + x\frac{dy}{dx} + y - e^y\frac{dy}{dx} = 0$.

Step:5.2

Factor out $\frac{dy}{dx}$ from the terms $x\frac{dy}{dx} - e^y\frac{dy}{dx}$.

Step:5.3

Separate the terms involving $\frac{dy}{dx}$ from those that do not.

Step:5.4

Solve for $\frac{dy}{dx}$ by dividing both sides by $(x - e^y)$, yielding $\frac{dy}{dx} = -\frac{3x^2 + y}{x - e^y}$.

Knowledge Notes:

1. Sum Rule: The derivative of a sum of functions is the sum of their derivatives.

2. Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.

3. Product Rule: The derivative of a product of two functions $f(x)$ and $g(x)$ is given by $f'(x)g(x) + f(x)g'(x)$.

4. Chain Rule: If a variable $y$ is a function of $u$ which is a function of $x$, then the derivative of $y$ with respect to $x$ is the derivative of $y$ with respect to $u$ multiplied by the derivative of $u$ with respect to $x$.

5. Exponential Rule: The derivative of $e^u$ with respect to $u$ is $e^u\ln(e)$, which simplifies to $e^u$ since $\ln(e) = 1$.

6. Implicit Differentiation: When a function is not given explicitly as $y=f(x)$ but is instead given in a form that involves both $x$ and $y$, we differentiate both sides of the equation with respect to $x$ and solve for $\frac{dy}{dx}$. This often involves using the chain rule for derivatives of $y$ with respect to $x$.